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Abstract algebraic logic : an introductory textbook / Josep Maria Font.

By: Font, Josep Maria, 1954-.
Material type: materialTypeLabelBookSeries: Studies in logic. Mathematical logic and foundations v. 60.Publisher: London : College Publications, c2016Description: xxix, 521 p. ; 24 cm.ISBN: 9781848902077.Other classification: 03-01 (03G27)
Contents:
Detailed contents -- Short contents vii -- Detailed contents ix -- A letter to the reader xv -- Introduction and Reading Guide xix -- Overview of the contents xxii -- Numbers, words, and symbols xxvi -- Further reading xxvii -- i Mathematical and logical preliminaries [1] -- 1.1 Sets, languages, algebras [1] -- The algebra of formulas [3] -- Evaluating the language into algebras [6] -- Equations and order relations [7] -- Sequents, and other wilder creatures [8] -- On variables and substitutions [9] -- Exercises for Section 1.1 [11] -- 1.2 Sentential logics [12] -- Examples: Syntactically defined logics [16] -- Examples: Semantically defined logics [18] -- What is a semantics? [22] -- What is an algebra-based semantics? [24] -- Soundness, adequacy, completeness [26] -- Extensions, fragments, expansions, reducts [27] -- Sentential-like notions of a logic on extended formulas [28] -- Exercises for Section 1.2 [30] -- 1.3 Closure operators and closure systems: the basics [32] -- Closure systems as ordered sets [37] -- Bases of a closure system [38] -- The family of all closure operators on a set [39] -- The Frege operator [41] -- Exercises for Section 1.3 [42] -- 1.4 Finitarity and structurality [44] -- Finitarity [45] -- Structurality [48] -- Exercises for Section 1.4 [52] -- 1.5 More on closure operators and closure systems [54] -- Lattices of closure operators and lattices of logics [54] -- Irreducible sets and saturated sets [55] -- Finitarity and compactness [57] -- Exercises for Section 1.5 [58] -- 1.6 Consequences associated with a class of algebras [59] -- The equational consequence, and varieties [60] -- The relative equational consequence [62] -- Quasivarieties and generalized quasivarieties 64 Relative congruences [65] -- The operator U [66] -- Exercises for SecHon 1.6 [67] -- 2 The first steps in the algebraic study of a logic [71] -- 2.1 From two-valued truth tables to Boolean algebras: the Lindenbaum-Tarski process for classical logic [71] -- Exercises for Section 2.1 [76] -- 2.2 Implicative logics [76] -- Exercises for Section 2.2 [86] -- 2.3 Filters [88] -- Hie general case [88] -- The implicative case [91] -- Exercises for Section 2.3 [96] -- 24 Extensions of the Lindenbaum-Tarski process [98] -- Implication-based extensions [98] -- Equivalence-based extensions [100] -- Conclusion [101] -- 2.5 Two digressions on first-order logic [102] -- The logic of the sentential connectives of first-order logic [102] -- The algebraic study of first-order logics [104] -- 3 The semantics of algebras [107] -- 3.1 Transformers, algebraic semantics, and assertional logics [108] -- Exercises for Section 3.1 [114] -- 3.2 Algebraizable logics [115] -- Uniqueness of the algebraization: the equivalent algebraic semantics [118] -- Exercises for Section [122] -- 3.3 A syntactic characterization, and the Lindenbaum-Tarski process again [123] -- Exercises for Section 3.3 [128] -- 3.4 More examples, and special kinds of algebraizable logics [129] -- Finitarity issues [132] -- Axiomatization [137] -- Regularly algebraizable logics [140] -- Exercises for Section 3.4 [143] -- 3.5 The Isomorphism Theorems [145] -- The evaluated transformers and their residuals [146] -- The theorems, in many versions [147] -- Regularity [155] -- Exercises for Section 3.5 [157] -- 3.6 Bridge theorems and transfer theorems [159] -- The classical Deduction Theorem [163] -- The general Deduction Theorem and its transfer [166] -- The Deduction Theorem in algebraizable logics and its applications [168] -- Weak versions of the Deduction Theorem [173] -- Exercises for Section 3.6 [175] -- 3.7 Generalizations and abstractions of algebraizability [176] -- Step 1: Algebraization of other sentential-like logical systems [176] -- Step 2: The notion of deductive equivalence [178] -- Step 3: Equivalence of structural closure operators [180] -- Step 4: Getting rid of points [181] -- 4 The semantics of matrices [183] -- 4.1 Logical matrices: basic concepts [183] -- Logics defined by matrices [184] -- Matrices as models of a logic [190] -- Exercises for Section 4.1 [192] -- 4.2 The Leibniz operator [194] -- Strict homomorphisms and the reduction process [199] -- Exercises for Section 4.2 [202] -- 4.3 Reduced models and Leibniz-reduced algebras [203] -- Exercises for Section 4.3 [212] -- 4.4 Applications to algebraizable logics [214] -- Exercises for Section 4.4 [220] -- 4.5 Matrices as relational structures [220] -- Model-theoretic characterizations [225] -- Exercises for Section 4.5 [233] -- 5 The semantics of generalized matrices [235] -- 5.1 Generalized matrices: basic concepts [236] -- Logics defined by generalized matrices [237] -- Generalized matrices as models of logics [240] -- Generalized matrices as models of Gentzen-style rules [242] -- Exercises for Section 5.1 [243] -- 5.2 Basic full generalized models, Tarski-style conditions and transfer theorems [244] -- Exercises for Section [251] -- 5.3 The Tarski operator and the Susako operator [254] -- Congruences in generalized matrices [254] -- Strict homomorphisms [259] -- Quotients [264] -- The process of reduction 266 Exercises for Section 5.3 [268] -- 5.4 The algebraic counterpart of a logic [270] -- The L-algebras and the intrinsic variety of a logic [273] -- Exercises for Section 5.4 [284] -- 5.5 Full generalized models [285] -- The main concept [285] -- Three case studies [288] -- The Isomorphism Theorem [294] -- The Galois adjunction of the compatibility relation [300] -- Exercises for Section 5.5 [302] -- 5.6 Generalized matrices as models of Gentzen systems [304] -- The notion of full adequacy [308] -- -- 6 Introduction to the Leibniz hierarchy -- 6.1 Overview [317] -- 6.2 Protoalgebraic logics [317] -- Basic concepts [322] -- The fundamental set and the syntactic characterization [323] -- Monotonicity, and its applications [327] -- A model-theoretic characterization [331] -- The Correspondence Theorem [333] -- Protoalgebraic logics and the Deduction Theorem [334] -- Full generalized models of protoalgebraic logics and Leibniz filters [341] -- Exercises for Section 6.2 [350] -- 6.3 Definability of equivalence (protoalgebraic and equivalential logics) [352] -- Definability of the Leibniz congruence, with or without parameters [352] -- Equivalential logics: Definition and general properties [557] -- Equivalentiality and properties of the Leibniz operator [360] -- Model-theoretic characterizations [362] -- Relation with relative equational consequences [367] -- Exercises for Section 6.3 [369] -- 6.4 Definability of truth (truth-equational, assertional and weakly algebraizable logics) [371] -- Implicit and explicit (equational) definitions of truth [373] -- Truth-equational logics [374] -- Assertional logics [383] -- Weakly algebraizable logics [387] -- Regularly weakly algebraizable logics [395] -- Exercises for Section 6.4 [397] -- 6.5 Algebraizable logics revisited [400] -- Full generalized models of algebraizable logics [404] -- On the bridge theorem concerning the Deduction Theorem [405] -- Regularly algebraizable logics revisited [407] -- Exercises for Section 6.5 [411] -- 7 Introduction to the Frege hierarchy [413] -- 7.1 Overview [413] -- 7.2 Selfextensional and fully selfextensional logics [419] -- Selfextensional logics with conjunction [421] -- Semilattice-based logics with an algebraizable assertional companion [434] -- Selfextensional logics with the Uniterm Deduction Theorem [440] -- Exercises for Sections 7.1 and 7.2 [446] -- 7.3 Fregean and fully Fregean logics [449] -- Fregean logics and truth-equational logics [455] -- Fregean logics and protoalgebraic logics [457] -- Exercises for Section 7.3 [467] -- Summary of properties of particular logics [471] -- Classical logic and its fragments [472] -- Intuitionistic logic, its fragments and extensions, and related logics [473] -- Other logics of implication [475] -- Modal logics [476] -- Many-valued logics [478] -- Substructural logics [480] -- Other logics [482] -- Ad hoc examples [483] -- Bibliography [485] -- Indices [497] -- Author index [497] -- Index of logics [500] -- Index of classes of algebras [503] -- General index [505] -- Index of acronyms and labels [512] -- Symbol index [515] --
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Incluye referencias bibliográficas (p. [485]-496) e índices.

Detailed contents --
Short contents vii --
Detailed contents ix --
A letter to the reader xv --
Introduction and Reading Guide xix --
Overview of the contents xxii --
Numbers, words, and symbols xxvi --
Further reading xxvii --
i Mathematical and logical preliminaries [1] --
1.1 Sets, languages, algebras [1] --
The algebra of formulas [3] --
Evaluating the language into algebras [6] --
Equations and order relations [7] --
Sequents, and other wilder creatures [8] --
On variables and substitutions [9] --
Exercises for Section 1.1 [11] --
1.2 Sentential logics [12] --
Examples: Syntactically defined logics [16] --
Examples: Semantically defined logics [18] --
What is a semantics? [22] --
What is an algebra-based semantics? [24] --
Soundness, adequacy, completeness [26] --
Extensions, fragments, expansions, reducts [27] --
Sentential-like notions of a logic on extended formulas [28] --
Exercises for Section 1.2 [30] --
1.3 Closure operators and closure systems: the basics [32] --
Closure systems as ordered sets [37] --
Bases of a closure system [38] --
The family of all closure operators on a set [39] --
The Frege operator [41] --
Exercises for Section 1.3 [42] --
1.4 Finitarity and structurality [44] --
Finitarity [45] --
Structurality [48] --
Exercises for Section 1.4 [52] --
1.5 More on closure operators and closure systems [54] --
Lattices of closure operators and lattices of logics [54] --
Irreducible sets and saturated sets [55] --
Finitarity and compactness [57] --
Exercises for Section 1.5 [58] --
1.6 Consequences associated with a class of algebras [59] --
The equational consequence, and varieties [60] --
The relative equational consequence [62] --
Quasivarieties and generalized quasivarieties 64 Relative congruences [65] --
The operator U [66] --
Exercises for SecHon 1.6 [67] --
2 The first steps in the algebraic study of a logic [71] --
2.1 From two-valued truth tables to Boolean algebras: the Lindenbaum-Tarski process for classical logic [71] --
Exercises for Section 2.1 [76] --
2.2 Implicative logics [76] --
Exercises for Section 2.2 [86] --
2.3 Filters [88] --
Hie general case [88] --
The implicative case [91] --
Exercises for Section 2.3 [96] --
24 Extensions of the Lindenbaum-Tarski process [98] --
Implication-based extensions [98] --
Equivalence-based extensions [100] --
Conclusion [101] --
2.5 Two digressions on first-order logic [102] --
The logic of the sentential connectives of first-order logic [102] --
The algebraic study of first-order logics [104] --
3 The semantics of algebras [107] --
3.1 Transformers, algebraic semantics, and assertional logics [108] --
Exercises for Section 3.1 [114] --
3.2 Algebraizable logics [115] --
Uniqueness of the algebraization: the equivalent algebraic semantics [118] --
Exercises for Section [122] --
3.3 A syntactic characterization, and the Lindenbaum-Tarski process again [123] --
Exercises for Section 3.3 [128] --
3.4 More examples, and special kinds of algebraizable logics [129] --
Finitarity issues [132] --
Axiomatization [137] --
Regularly algebraizable logics [140] --
Exercises for Section 3.4 [143] --
3.5 The Isomorphism Theorems [145] --
The evaluated transformers and their residuals [146] --
The theorems, in many versions [147] --
Regularity [155] --
Exercises for Section 3.5 [157] --
3.6 Bridge theorems and transfer theorems [159] --
The classical Deduction Theorem [163] --
The general Deduction Theorem and its transfer [166] --
The Deduction Theorem in algebraizable logics and its applications [168] --
Weak versions of the Deduction Theorem [173] --
Exercises for Section 3.6 [175] --
3.7 Generalizations and abstractions of algebraizability [176] --
Step 1: Algebraization of other sentential-like logical systems [176] --
Step 2: The notion of deductive equivalence [178] --
Step 3: Equivalence of structural closure operators [180] --
Step 4: Getting rid of points [181] --
4 The semantics of matrices [183] --
4.1 Logical matrices: basic concepts [183] --
Logics defined by matrices [184] --
Matrices as models of a logic [190] --
Exercises for Section 4.1 [192] --
4.2 The Leibniz operator [194] --
Strict homomorphisms and the reduction process [199] --
Exercises for Section 4.2 [202] --
4.3 Reduced models and Leibniz-reduced algebras [203] --
Exercises for Section 4.3 [212] --
4.4 Applications to algebraizable logics [214] --
Exercises for Section 4.4 [220] --
4.5 Matrices as relational structures [220] --
Model-theoretic characterizations [225] --
Exercises for Section 4.5 [233] --
5 The semantics of generalized matrices [235] --
5.1 Generalized matrices: basic concepts [236] --
Logics defined by generalized matrices [237] --
Generalized matrices as models of logics [240] --
Generalized matrices as models of Gentzen-style rules [242] --
Exercises for Section 5.1 [243] --
5.2 Basic full generalized models, Tarski-style conditions and transfer theorems [244] --
Exercises for Section [251] --
5.3 The Tarski operator and the Susako operator [254] --
Congruences in generalized matrices [254] --
Strict homomorphisms [259] --
Quotients [264] --
The process of reduction 266 Exercises for Section 5.3 [268] --
5.4 The algebraic counterpart of a logic [270] --
The L-algebras and the intrinsic variety of a logic [273] --
Exercises for Section 5.4 [284] --
5.5 Full generalized models [285] --
The main concept [285] --
Three case studies [288] --
The Isomorphism Theorem [294] --
The Galois adjunction of the compatibility relation [300] --
Exercises for Section 5.5 [302] --
5.6 Generalized matrices as models of Gentzen systems [304] --
The notion of full adequacy [308] --
--
6 Introduction to the Leibniz hierarchy --
6.1 Overview [317] --
6.2 Protoalgebraic logics [317] --
Basic concepts [322] --
The fundamental set and the syntactic characterization [323] --
Monotonicity, and its applications [327] --
A model-theoretic characterization [331] --
The Correspondence Theorem [333] --
Protoalgebraic logics and the Deduction Theorem [334] --
Full generalized models of protoalgebraic logics and Leibniz filters [341] --
Exercises for Section 6.2 [350] --
6.3 Definability of equivalence (protoalgebraic and equivalential logics) [352] --
Definability of the Leibniz congruence, with or without parameters [352] --
Equivalential logics: Definition and general properties [557] --
Equivalentiality and properties of the Leibniz operator [360] --
Model-theoretic characterizations [362] --
Relation with relative equational consequences [367] --
Exercises for Section 6.3 [369] --
6.4 Definability of truth (truth-equational, assertional and weakly algebraizable logics) [371] --
Implicit and explicit (equational) definitions of truth [373] --
Truth-equational logics [374] --
Assertional logics [383] --
Weakly algebraizable logics [387] --
Regularly weakly algebraizable logics [395] --
Exercises for Section 6.4 [397] --
6.5 Algebraizable logics revisited [400] --
Full generalized models of algebraizable logics [404] --
On the bridge theorem concerning the Deduction Theorem [405] --
Regularly algebraizable logics revisited [407] --
Exercises for Section 6.5 [411] --
7 Introduction to the Frege hierarchy [413] --
7.1 Overview [413] --
7.2 Selfextensional and fully selfextensional logics [419] --
Selfextensional logics with conjunction [421] --
Semilattice-based logics with an algebraizable assertional companion [434] --
Selfextensional logics with the Uniterm Deduction Theorem [440] --
Exercises for Sections 7.1 and 7.2 [446] --
7.3 Fregean and fully Fregean logics [449] --
Fregean logics and truth-equational logics [455] --
Fregean logics and protoalgebraic logics [457] --
Exercises for Section 7.3 [467] --
Summary of properties of particular logics [471] --
Classical logic and its fragments [472] --
Intuitionistic logic, its fragments and extensions, and related logics [473] --
Other logics of implication [475] --
Modal logics [476] --
Many-valued logics [478] --
Substructural logics [480] --
Other logics [482] --
Ad hoc examples [483] --
Bibliography [485] --
Indices [497] --
Author index [497] --
Index of logics [500] --
Index of classes of algebras [503] --
General index [505] --
Index of acronyms and labels [512] --
Symbol index [515] --

MR, MR3558731

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